对于一个紧的、连通的、单连通的李群G，环群LG是由H^1-Sobolev映射S^1-->G组成的无限维Hilbert李群。LG及其齐性空间的几何与表示论有关，并已被广泛研究。基环空间Omega（G）是$LG$的齐性空间的一个例子，并且具有自然的哈密顿量T x S^1作用，其中T是G的极大环面。我们研究了这个动作的矩映射亩，特别是证明了它的定期水平集连接。该结果是Atiyah定理的无限维模拟，该定理指出紧致辛流形上汉密尔顿环面作用的矩映射的原像是连通的。在有限维的情况下，这个连通性的结果被用来证明的矩映射的图像为一个紧凑的哈密顿T-空间是凸的。因此，我们的定理也可以被看作是一个同伴的结果阿蒂亚和普雷斯利定理，其中规定，图像μ（Ω（G））是凸的。我们还证明了对于能量泛函E，即S^1旋转作用量的矩映射，每个非空原像是连通的。
英文摘要：
For a compact, connected, simply-connected Lie group G, the loop group LG is the infinite-dimensional Hilbert Lie group consisting of H^1-Sobolev maps S^1-->G. The geometry of LG and its homogeneous spaces is related to representation theory and has been extensively studied. The space of based loops Omega(G) is an example of a homogeneous space of $LG$ and has a natural Hamiltonian T x S^1 action, where T is the maximal torus of G. We study the moment map mu for this action, and in particular prove that its regular level sets are connected. This result is as an infinite-dimensional analogue of a theorem of Atiyah that states that the preimage of a moment map for a Hamiltonian torus action on a compact symplectic manifold is connected. In the finite-dimensional case, this connectivity result is used to prove that the image of the moment map for a compact Hamiltonian T-space is convex. Thus our theorem can also be viewed as a companion result to a theorem of Atiyah and Pressley, which states that the image mu(Omega(G)) is convex. We also show that for the energy functional E, which is the moment map for the S^1 rotation action, each non-empty preimage is connected.
The Ricci form is a moment map for the action of the group of exact volume preserving diffeomorphisms on the space of almost complex structures. This observation yields a new approach to the Weil-Petersson symplectic form on the Teichmuller space of isotopy classes of complex structures with real first Chern class zero and nonempty Kahler cone.
Unimodality of the Betti numbers for Hamiltonian circle actions with index-increasing moment maps
Tolman在2005年“Moment maps in Various Geometry”会议上提出的单峰猜想指出，如果（M，w）是一个2n维光滑紧致辛流形，具有一个Hamilton圈作用，且仅有孤立不动点，则贝蒂数序列是单峰的.最近，作者和M.金证明了单峰持有八维情况下使用等变上同调理论。本文将\cite{CK}中的思想推广到任意维的情形。在任意维数下证明了该猜想，并假设一个矩映射是指数递增的。"
